B. Exponential Functions · Financial Mathematics: Compound Interest and Annuities

Finding the Interest Rate: Rational Exponents and the TVM Solver

B3.4About 70 minutes

Learning goals

I can find the interest rate per compounding period, i, when P, A and n are known.
I can solve (1 + i)ⁿ = A ÷ P using a rational exponent on a scientific calculator.
I can use a TVM Solver (on a graphing calculator or a website) or a spreadsheet RATE function to find the rate.
I can convert the rate per period to an annual rate and use it to compare investments or loans.

Words to know

rational exponent
An exponent that is a fraction. x^(1/n) is the nth root of x, so it undoes raising to the power n.

TVM Solver
A time value of money tool, found on graphing calculators and many websites. Enter all values but one and it solves for the missing one.

nominal annual rate
The stated yearly rate, such as "6% compounded monthly". It equals i × m.

cash-flow sign convention
In a TVM Solver, money you pay out is entered as negative and money you receive is positive.

appreciation
An increase in value over time, such as a collectible or a home increasing in price.

What rate did that grow at?

Alex's parents bought a rare comic book for $2 000 four years ago. Today it is worth $2 500. What annual rate of growth, compounded annually, does that represent? This time you know P, A and n. The unknown is i, which sits inside the base of the power. To find it you will undo the exponent.

Method 1: Rational exponents (scientific calculator)

Start with A = P(1 + i)ⁿ.

Divide both sides by P: (1 + i)ⁿ = A ÷ P
Raise both sides to the power 1/n (this takes the nth root): 1 + i = (A ÷ P)^(1/n)
Subtract 1: i = (A ÷ P)^(1/n) − 1

This works because (xⁿ)^(1/n) = x, an exponent law from the unit Exponential Functions.

On a calculator, type (2500 ÷ 2000) ^ (1 ÷ 4) − 1. Use brackets around 1 ÷ 4; without them the calculator computes (2500 ÷ 2000)¹ ÷ 4.

Worked example

Example 1: Annual compounding

$2 000 grew to $2 500 in 4 years, compounded annually. Find the annual interest rate.

(1 + i)⁴ = 2500 ÷ 2000 = 1.25
1 + i = 1.25^(1/4) ≈ 1.0573713
i ≈ 0.0573713, or about 5.74%
Check: 2000(1.0574)⁴ ≈ 2500.27, close to 2500 (the small difference comes from rounding i). ✓

The value grew at about 5.74% per year, compounded annually.

Worked example

Example 2: Monthly compounding

An investment of $5 000 is worth $6 000 after 3 years, with interest compounded monthly. Find i (per month) and the nominal annual rate.

n = 3 × 12 = 36.
(1 + i)³⁶ = 6000 ÷ 5000 = 1.2
i = 1.2^(1/36) − 1 ≈ 0.0050773 per month.
Annual rate = 12 × i ≈ 0.060928, or about 6.09% compounded monthly.

i ≈ 0.5077% per month, so the rate is about 6.09% per year, compounded monthly.

A graphing approach to Example 1: graph the amount after 4 years for different rates, and find where the curve meets the line A = 2500. The intersection is at i ≈ 0.0574, or 5.74%. (The vertical axis starts at $2 000.)

Method 2: The TVM Solver

Many graphing calculators have a TVM Solver (on TI-83/84 calculators it is in the Finance app, APPS → Finance → TVM Solver), and many free websites offer the same tool. You enter every value except the one you want, move the cursor to the unknown, and press SOLVE (ALPHA ENTER on a TI-84).

N: total number of compounding periods
I%: the annual rate as a percent (not a decimal, not per period)
PV: present value
PMT: regular payment (0 when there is a single lump sum)
FV: future value
P/Y and C/Y: payments per year and compounding periods per year (for a lump sum, set both equal to the compounding frequency)

Signs matter. Money that leaves your hands is negative; money you receive is positive. If you invest $5 000, enter PV = −5000; you get back FV = 6000. If PV and FV have the same sign, the solver gives an error, because you cannot pay out and also get nothing back.

Example 2 in a TVM Solver. The investment of $5 000 is entered as −5000 (money paid out). Solving for I% gives 6.09%, the nominal annual rate compounded monthly.

Method 3: Spreadsheets

In Google Sheets, Excel or LibreOffice, =RATE(nper, pmt, pv, fv) gives the rate per period. For Example 2: =RATE(36, 0, -5000, 6000) gives about 0.0050773. Multiply by 12 for the annual rate. The same sign convention applies.

Worked example

Example 3: Finding the rate on a debt

A "pay later" plan lets Priya take home a $1 200 laptop now and pay $1 452 in a single payment 2 years later. If interest is compounded monthly, what annual rate is she really paying?

P = 1200, A = 1452, n = 24.
(1 + i)²⁴ = 1452 ÷ 1200 = 1.21
i = 1.21^(1/24) − 1 ≈ 0.0079741 per month.
Annual rate ≈ 12 × 0.0079741 ≈ 0.0957, or about 9.57% compounded monthly.
TVM check: N = 24, PV = 1200 (she receives the laptop's value), PMT = 0, FV = −1452 (she pays), P/Y = C/Y = 12 → I% ≈ 9.57.

Priya is paying about 9.57% per year compounded monthly, even though the plan never mentions a rate.

Just for fun

A TVM Solver walks into a bank and asks for a loan. The banker says, "Fill in everything except one box." The solver says, "Finally, a form I understand."

Hands-on activity

Track a price you care about

You need: a computer or phone with a calculator (and, if possible, a free online TVM calculator or spreadsheet), and paper.

Choose something whose price you can find for two different years: a movie ticket, a litre of milk, a postage stamp, a used car model, or a sports ticket. Old flyers, receipts, family memories or a reliable website can help. Write down both prices, both years, and where you found them.
Treat the older price as P and the newer price as A. Let n be the number of years between them.
Find the average annual growth rate, compounded annually, with i = (A ÷ P)^(1/n) − 1.
Check your answer with a TVM Solver or =RATE(n, 0, -P, A).
Compare with the Bank of Canada's inflation-control target of 2% per year (the target in place in 2026). Did your item grow faster or slower than that?

What to look for: correct n (years between the prices), brackets around 1/n, a percent rounded to two decimals, and a sensible comparison sentence.

Your turn

Solve for i: (1 + i)⁵ = 1.25. Give i as a percent to two decimals.

$2 400 grows to $3 000 in 5 years, compounded annually. Find the annual rate.

$1 500 grows to $1 800 in 2 years, compounded quarterly. Find i per quarter and the annual rate.

$8 000 grows to $10 000 in 3 years, compounded semi-annually. Find the annual rate.

Write the TVM Solver entries to find the rate in the last question.

Why does the TVM Solver give an error if you enter PV = 8000 and FV = 10 000 (both positive)?

More practice: warm-up

Evaluate 16^(1/4).

Evaluate 1.21^(1/2).

If 1 + i = 1.035, what is i as a percent?

i = 0.0075 per month. What is the nominal annual rate?

In a TVM Solver, is the I% entry the annual rate or the rate per period?

More practice: on your own

Money doubles in 10 years, compounded annually. What is the rate?

A $500 deposit is worth $650 after 3 years, compounded monthly. Find the nominal annual rate.

A loan of $4 000 with no payments grows to $5 000 in 5 years, compounded quarterly. What rate is charged?

A hockey card bought for $9 000 sold for $15 000 seven years later. What annual rate of appreciation is that (compounded annually)?

Investment A turns $750 into $900 in 3 years (compounded annually). Investment B turns $1 000 into $1 500 in 10 years (compounded annually). Which has the higher rate?

Challenge

At what annual rate, compounded annually, would money triple in 15 years?

A $2 000 loan grows to $2 600 in 2 years with no payments, compounded monthly. Is this rate above or below the Criminal Code limit of 35% APR that has applied in Canada since January 1, 2025? Show your work.

Jade says, "$1 000 became $1 210 in 2 years, so the rate is 21% ÷ 2 = 10.5% per year." Explain her mistake and find the correct rate, compounded annually.

Check your understanding

Quiz

Which expression gives i if (1 + i)⁸ = 1.4?

$1 000 grows to $1 102.50 in 2 years, compounded annually. The rate is:

In a TVM Solver, if you invest $3 000 today, you should enter:

The rate per month is 0.4%. What is the nominal annual rate?

To find the annual rate on a TI-84 TVM Solver for monthly compounding with no payments, set:

Money doubles in 6 years, compounded annually. The rate is closest to:

For parents and teachers

Big idea (B3.4): Students solve compound interest problems for the interest rate per compounding period, i, using a TVM Solver on a graphing calculator or website. On this site we also show the rational exponent method, i = (A ÷ P)^(1/n) − 1, so every answer can be found with a scientific calculator. The next lesson finds the number of periods, n.

How to help: If you have a free online TVM or compound interest calculator, let your teen show you how to fill it in, and ask them to explain why one of PV or FV must be negative. The activity uses real prices from two different years, a good chance to share memories of what things used to cost.

Common difficulties:

Missing brackets around 1/n on the calculator.
Reporting i per month as the annual rate (forgetting to multiply by 12).
Entering I% as a decimal (0.06) instead of a percent (6) in a TVM Solver.
Using simple interest thinking (dividing the total growth by the years).

Canadian facts used (checked October 2026): the Bank of Canada's inflation-control target is 2% (bankofcanada.ca); Canada's criminal interest rate is 35% APR, in force since January 1, 2025 (Canada Gazette, Part II, June 19, 2024).